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Restrictions of Unitary Representations of Real Reductive Groups Restrictions of Unitary Representations of Real Reductive Groups Toshiyuki Kobayashi Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan [email protected] 1 Reductive Lie groups Classical groups such as the general linear group GL(n, R) and the Lorentz group O(p, q) are reductive Lie groups. This section tries to give an elementary introduc- tion to the structures of reductive Lie groups based on examples, which will be used throughout this chapter. All the materials of this section and further details may be found in standard textbooks or lecture notes such as [23, 36, 99, 104]. 1.1 Smallest objects The “smallest objects” of representations are irreducible representations. All uni- tary representations are built up of irreducible unitary representations by means of direct integrals (see §3.1.2). The “smallest objects” for Lie groups are those without nontrivial connected normal subgroups; they consist of simple Lie groups such as SL(n, R), and one- dimensional abelian Lie groups such as R and S1. Reductive Lie groups are locally isomorphic to these Lie groups or their direct products. Loosely, a theorem of Duflo [10] asserts that all irreducible unitary representa- tions of a real algebraic group are built from those of reductive Lie groups. Throughout this chapter, our main concern will be with irreducible decomposi- tions of unitary representations of reductive Lie groups. In Section 1 we summarize necessary notation and basic facts on reductive Lie groups in as elementary a way as possible. Keywords and phrases: unitary representations, branching laws, semisimple Lie group, dis- crete spectrum, homogeneous space 2000 MSC: primary 22E4; secondary 43A85, 11F67, 53C50, 53D20 140 T. Kobayashi 1.2 Generallinear group GL(N, R) The general linear group GL(N, R) is a typical example of reductive Lie groups. First, we set up notation for G := GL(N, R). We consider the map − θ : G → G, g → tg 1. Clearly, θ is an involutive automorphism in the sense that θ satisfies: θ ◦ θ = id, θ is an automorphism of the Lie group G. The set of the fixed points of θ θ K := G ={g ∈ G : θg = g} − ={g ∈ GL(N, R) : tg 1 = g} is nothing but the orthogonal group O(N), which is compact because O(N) is a 2 bounded closed set in the Euclidean space in M(N, R) RN in light of the follow- ing inclusion: N N ( ) ⊂{ = ( ) ∈ ( , R) 2 = }. O N g gij M N : gij N i=1 j=1 Furthermore, there is no larger compact subgroupofGL(N, R) which contains O(N).Thus, K = O(N) is a maximal compact subgroup of G = GL(N, R). Conversely, any maximal compact subgroupofG is conjugate to K by an element of G. The involution θ (or its conjugation) is called a Cartaninvolution of GL(N, R). 1.3 Cartan decomposition The Lie algebras of G and K will be denoted by g and k, respectively. Then, for G = GL(N, R) and K = O(N), we have the following direct sum decomposition: g = gl(N, R) := the set of N × N real matrices % k = o(N) :={X ∈ gl(N, R) : X =−tX} ⊕ p = Symm(N, R) :={X ∈ gl(N, R) : X = tX}. The decomposition g = k + p is called the Cartan decomposition of the Lie algebra g corresponding to the Cartan involution θ. This decomposition lifts to the Lie group G in the sense that we have the diffeomorphism Restrictions of Unitary Representations of Real Reductive Groups 141 ∼ p × K → G,(X, k) → eX k. (1.3.1) The map (1.3.1) is bijective, as (X, k) is recovered from g ∈ GL(N, R) by the formula 1 − X = log(gtg), k = e X g. 2 Here, log is the inverse of the bijection exp : Symm(N, R) → Symm+(N, R) :={X ∈ Symm(N, R) : X 0}. It requires a small computation of the Jacobian to see the map (1.3.1) is a Cω- diffeomorphism (see [23, Chapter II, Theorem 1.7]). The decomposition (1.3.1) is known as the polar decomposition of GL(N, R) in linear algebra, and is a special case of a Cartan decomposition of a reductive Lie group in Lie theory (see (1.4.2) below). 1.4 Reductive Lie groups A connected Lie group G is called reductive if its Lie algebra g is reductive, namely, it is isomorphic to the direct sum of simple Lie algebras and an abelian Lie algebra. In the literature on representation theory, however, different authors have introduced and/or adopted several variations of the category of reductive Lie groups, in particu- lar, with respect to discreteness, linearity and covering. In this chapter we adopt the following definition of “reductive Lie group”. Some advantages here are: • The structural theory of reductive Lie group can be explained in an elementary way, parallel to that of GL(n, R). • It is easy to verify that (typical) classical groups are indeed real reductive Lie groups (see §1.5). Definition 1.4. We say a Lie group G is linearreductive if G can be realized as a closed subgroupofGL(N, R) satisfying the following two conditions: i) θG = G. ii) G has at most finitely many connected components. In this chapter we say G is a reductive Lie group if it is a finite covering of a linear reductive Lie group. Then the condition (i) of Definition 1.4 implies that its Lie algebra g is reductive. If G is linear reductive, then by using its realization in GL(N, R) we define K := G ∩ O(N). Then K is a maximal compact subgroupofG. The Lie algebra k of K is given by g ∩ o(N). We define p := g ∩ Symm(N, R). Similar to the case of GL(N, R), we have the following Cartan decompositions: 142 T. Kobayashi g = k + p (direct sum decomposition) (1.4.1) ∼ p × K → G (diffeomorphism) (1.4.2) by taking the restriction of the corresponding decompositions for GL(N, R) (see (1.3.1)) to g and G, respectively. 1.5 Examples of reductive Lie groups Reductive Lie groups in the above definition include all compact Lie groups (espe- cially finite groups) and classical groups such as the general linear group GL(N, R), GL(N, C), the special linear group SL(N, R), SL(N, C), the generalized Lorentz group (the indefinite orthogonal group) O(p, q), the symplectic group Sp(N, R), Sp(N, C), and some others such as U(p, q), Sp(p, q), U ∗(2n), O∗(2n), etc. In this subsection, we review some of these classical groups, and explain how to prove that they are reductive Lie groups in the sense of Definition 1.4. For this purpose, the following theorem is useful. Theorem 1.5.1. Let G be a linear algebraic subgroupofGL(N, R).IfθG = G, thenGis a reductive Lie group. Here we recall that a linear algebraic group (over R)isasubgroup G ⊂ GL(N, R) which is an algebraic subset in M(N, R), that is, the set of zeros of an ideal of polynomial functions with coefficients in R. Sketch of proof. In order to prove that G has at most finitely many connected compo- nents, it is enough to verify the following two assertions: • K = G ∩ O(N) is compact. ∼ • The map p × K → G,(X, k) → eX k is a homeomorphism (Cartan decompo- sition). The first assertion is obvious because G is closed. The second assertion is deduced from the bijection (1.3.1) for GL(N, R). For this, the nontrivial part is a proof of the implication “eX k ∈ G ⇒ X ∈ p”. Let us prove this. We note that if eX k ∈ G, then e2X = (eX k)(θ(eX k))−1 ∈ G, and therefore e2nX ∈ G for all n ∈ Z.ToseeX ∈ p, we want to show e2tX ∈ G for all t ∈ R. This follows from Chevalley’s lemma: let X ∈ Symm(N, R);ifesX satisfies a polynomial equation of entries for any s ∈ Z, then so does esX for any s ∈ R. As special cases of Theorem 1.5.1, we pin down Propositions 1.5.2, 1.5.3 and 1.5.6. Proposition 1.5.2. SL(N, R) :={g ∈ GL(N, R) : det g = 1} is a reductive Lie group. Proof. Obvious from Theorem 1.5.1. Restrictions of Unitary Representations of Real Reductive Groups 143 Proposition 1.5.3. Let A be an N × N matrix such thatA2 = cI (c = 0).Then G(A) :={g ∈ GL(N, R) : tg Ag = A} is a reductive Lie group. Proof. First, it follows from the definition that G(A) is a linear algebraic subgroup of GL(N, R). Second, let us prove θ(G(A)) = G(A).Iftg Ag = A, then g−1 = A−1tg A. Since −1 = 1 A c A,wehave 1 I = gg−1 = g( A)tg(cA−1) = g Atg A−1. c Hence g Atg = A, that is θg ∈ G(A). Then the proposition follows from Theo- rem 1.5.1. I O Example 1.5.4. If A = p (p + q = n), then O −Iq G(A) = O(p, q)(indefinite orthogonal group). In this case K = O(p, q) ∩ O(p + q) O(p) × O(q). OB p ={ : B ∈ M(p, q; R)}. tBO OI Example 1.5.5. If A = n (N = 2n), then −In O G(A) = Sp(n, R) (real symplectic group). In this case, K = Sp(n, R) ∩ O(2n) U(n). AB ∼ k ={ : A =−tA, B = tB} → u(n), −BA AB → A + iB.
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